Theorem ragperp 28725

Description: Deduce that two lines are perpendicular from a right angle statement. One direction of theorem 8.13 of [Schwabhauser] p. 59. (Contributed by Thierry Arnoux, 20-Oct-2019.)

Hypotheses

Ref	Expression
isperp.p	⊢ 𝑃 = (Base‘𝐺)
isperp.d	⊢ − = (dist‘𝐺)
isperp.i	⊢ 𝐼 = (Itv‘𝐺)
isperp.l	⊢ 𝐿 = (LineG‘𝐺)
isperp.g	⊢ (𝜑 → 𝐺 ∈ TarskiG)
isperp.a	⊢ (𝜑 → 𝐴 ∈ ran 𝐿)
ragperp.b	⊢ (𝜑 → 𝐵 ∈ ran 𝐿)
ragperp.x	⊢ (𝜑 → 𝑋 ∈ (𝐴 ∩ 𝐵))
ragperp.u	⊢ (𝜑 → 𝑈 ∈ 𝐴)
ragperp.v	⊢ (𝜑 → 𝑉 ∈ 𝐵)
ragperp.1	⊢ (𝜑 → 𝑈 ≠ 𝑋)
ragperp.2	⊢ (𝜑 → 𝑉 ≠ 𝑋)
ragperp.r	⊢ (𝜑 → ⟨“𝑈𝑋𝑉”⟩ ∈ (∟G‘𝐺))

Assertion

Ref	Expression
ragperp	⊢ (𝜑 → 𝐴(⟂G‘𝐺)𝐵)

Proof of Theorem ragperp

Dummy variables 𝑢 𝑣 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		isperp.p	. . . 4 ⊢ 𝑃 = (Base‘𝐺)
2		isperp.d	. . . 4 ⊢ − = (dist‘𝐺)
3		isperp.i	. . . 4 ⊢ 𝐼 = (Itv‘𝐺)
4		isperp.l	. . . 4 ⊢ 𝐿 = (LineG‘𝐺)
5		eqid 2737	. . . 4 ⊢ (pInvG‘𝐺) = (pInvG‘𝐺)
6		isperp.g	. . . . 5 ⊢ (𝜑 → 𝐺 ∈ TarskiG)
7	6	adantr 480	. . . 4 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐵)) → 𝐺 ∈ TarskiG)
8		ragperp.b	. . . . . 6 ⊢ (𝜑 → 𝐵 ∈ ran 𝐿)
9	8	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐵)) → 𝐵 ∈ ran 𝐿)
10		simprr 773	. . . . 5 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐵)) → 𝑣 ∈ 𝐵)
11	1, 4, 3, 7, 9, 10	tglnpt 28557	. . . 4 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐵)) → 𝑣 ∈ 𝑃)
12		isperp.a	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ ran 𝐿)
13	12	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐵)) → 𝐴 ∈ ran 𝐿)
14		ragperp.x	. . . . . . 7 ⊢ (𝜑 → 𝑋 ∈ (𝐴 ∩ 𝐵))
15	14	elin1d 4204	. . . . . 6 ⊢ (𝜑 → 𝑋 ∈ 𝐴)
16	15	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐵)) → 𝑋 ∈ 𝐴)
17	1, 4, 3, 7, 13, 16	tglnpt 28557	. . . 4 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐵)) → 𝑋 ∈ 𝑃)
18		simprl 771	. . . . 5 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐵)) → 𝑢 ∈ 𝐴)
19	1, 4, 3, 7, 13, 18	tglnpt 28557	. . . 4 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐵)) → 𝑢 ∈ 𝑃)
20		ragperp.v	. . . . . . 7 ⊢ (𝜑 → 𝑉 ∈ 𝐵)
21	20	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐵)) → 𝑉 ∈ 𝐵)
22	1, 4, 3, 7, 9, 21	tglnpt 28557	. . . . 5 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐵)) → 𝑉 ∈ 𝑃)
23		ragperp.u	. . . . . . . . 9 ⊢ (𝜑 → 𝑈 ∈ 𝐴)
24	23	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐵)) → 𝑈 ∈ 𝐴)
25	1, 4, 3, 7, 13, 24	tglnpt 28557	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐵)) → 𝑈 ∈ 𝑃)
26		ragperp.r	. . . . . . . 8 ⊢ (𝜑 → ⟨“𝑈𝑋𝑉”⟩ ∈ (∟G‘𝐺))
27	26	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐵)) → ⟨“𝑈𝑋𝑉”⟩ ∈ (∟G‘𝐺))
28		ragperp.1	. . . . . . . 8 ⊢ (𝜑 → 𝑈 ≠ 𝑋)
29	28	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐵)) → 𝑈 ≠ 𝑋)
30	23	ad2antrr 726	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐵)) ∧ ¬ 𝑋 = 𝑢) → 𝑈 ∈ 𝐴)
31	6	ad2antrr 726	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐵)) ∧ ¬ 𝑋 = 𝑢) → 𝐺 ∈ TarskiG)
32	17	adantr 480	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐵)) ∧ ¬ 𝑋 = 𝑢) → 𝑋 ∈ 𝑃)
33	19	adantr 480	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐵)) ∧ ¬ 𝑋 = 𝑢) → 𝑢 ∈ 𝑃)
34		simpr 484	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐵)) ∧ ¬ 𝑋 = 𝑢) → ¬ 𝑋 = 𝑢)
35	34	neqned 2947	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐵)) ∧ ¬ 𝑋 = 𝑢) → 𝑋 ≠ 𝑢)
36	12	ad2antrr 726	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐵)) ∧ ¬ 𝑋 = 𝑢) → 𝐴 ∈ ran 𝐿)
37	15	ad2antrr 726	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐵)) ∧ ¬ 𝑋 = 𝑢) → 𝑋 ∈ 𝐴)
38	18	adantr 480	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐵)) ∧ ¬ 𝑋 = 𝑢) → 𝑢 ∈ 𝐴)
39	1, 3, 4, 31, 32, 33, 35, 35, 36, 37, 38	tglinethru 28644	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐵)) ∧ ¬ 𝑋 = 𝑢) → 𝐴 = (𝑋𝐿𝑢))
40	30, 39	eleqtrd 2843	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐵)) ∧ ¬ 𝑋 = 𝑢) → 𝑈 ∈ (𝑋𝐿𝑢))
41	40	ex 412	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐵)) → (¬ 𝑋 = 𝑢 → 𝑈 ∈ (𝑋𝐿𝑢)))
42	41	orrd 864	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐵)) → (𝑋 = 𝑢 ∨ 𝑈 ∈ (𝑋𝐿𝑢)))
43	42	orcomd 872	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐵)) → (𝑈 ∈ (𝑋𝐿𝑢) ∨ 𝑋 = 𝑢))
44	1, 2, 3, 4, 5, 7, 25, 17, 22, 19, 27, 29, 43	ragcol 28707	. . . . . 6 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐵)) → ⟨“𝑢𝑋𝑉”⟩ ∈ (∟G‘𝐺))
45	1, 2, 3, 4, 5, 7, 19, 17, 22, 44	ragcom 28706	. . . . 5 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐵)) → ⟨“𝑉𝑋𝑢”⟩ ∈ (∟G‘𝐺))
46		ragperp.2	. . . . . 6 ⊢ (𝜑 → 𝑉 ≠ 𝑋)
47	46	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐵)) → 𝑉 ≠ 𝑋)
48	20	ad2antrr 726	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐵)) ∧ ¬ 𝑋 = 𝑣) → 𝑉 ∈ 𝐵)
49	6	ad2antrr 726	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐵)) ∧ ¬ 𝑋 = 𝑣) → 𝐺 ∈ TarskiG)
50	17	adantr 480	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐵)) ∧ ¬ 𝑋 = 𝑣) → 𝑋 ∈ 𝑃)
51	11	adantr 480	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐵)) ∧ ¬ 𝑋 = 𝑣) → 𝑣 ∈ 𝑃)
52		simpr 484	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐵)) ∧ ¬ 𝑋 = 𝑣) → ¬ 𝑋 = 𝑣)
53	52	neqned 2947	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐵)) ∧ ¬ 𝑋 = 𝑣) → 𝑋 ≠ 𝑣)
54	8	ad2antrr 726	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐵)) ∧ ¬ 𝑋 = 𝑣) → 𝐵 ∈ ran 𝐿)
55	14	elin2d 4205	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
56	55	ad2antrr 726	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐵)) ∧ ¬ 𝑋 = 𝑣) → 𝑋 ∈ 𝐵)
57	10	adantr 480	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐵)) ∧ ¬ 𝑋 = 𝑣) → 𝑣 ∈ 𝐵)
58	1, 3, 4, 49, 50, 51, 53, 53, 54, 56, 57	tglinethru 28644	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐵)) ∧ ¬ 𝑋 = 𝑣) → 𝐵 = (𝑋𝐿𝑣))
59	48, 58	eleqtrd 2843	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐵)) ∧ ¬ 𝑋 = 𝑣) → 𝑉 ∈ (𝑋𝐿𝑣))
60	59	ex 412	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐵)) → (¬ 𝑋 = 𝑣 → 𝑉 ∈ (𝑋𝐿𝑣)))
61	60	orrd 864	. . . . . 6 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐵)) → (𝑋 = 𝑣 ∨ 𝑉 ∈ (𝑋𝐿𝑣)))
62	61	orcomd 872	. . . . 5 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐵)) → (𝑉 ∈ (𝑋𝐿𝑣) ∨ 𝑋 = 𝑣))
63	1, 2, 3, 4, 5, 7, 22, 17, 19, 11, 45, 47, 62	ragcol 28707	. . . 4 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐵)) → ⟨“𝑣𝑋𝑢”⟩ ∈ (∟G‘𝐺))
64	1, 2, 3, 4, 5, 7, 11, 17, 19, 63	ragcom 28706	. . 3 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐵)) → ⟨“𝑢𝑋𝑣”⟩ ∈ (∟G‘𝐺))
65	64	ralrimivva 3202	. 2 ⊢ (𝜑 → ∀𝑢 ∈ 𝐴 ∀𝑣 ∈ 𝐵 ⟨“𝑢𝑋𝑣”⟩ ∈ (∟G‘𝐺))
66	1, 2, 3, 4, 6, 12, 8, 14	isperp2 28723	. 2 ⊢ (𝜑 → (𝐴(⟂G‘𝐺)𝐵 ↔ ∀𝑢 ∈ 𝐴 ∀𝑣 ∈ 𝐵 ⟨“𝑢𝑋𝑣”⟩ ∈ (∟G‘𝐺)))
67	65, 66	mpbird 257	1 ⊢ (𝜑 → 𝐴(⟂G‘𝐺)𝐵)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2108   ≠ wne 2940  ∀wral 3061   ∩ cin 3950   class class class wbr 5143  ran crn 5686  ‘cfv 6561  (class class class)co 7431  ⟨“cs3 14881  Basecbs 17247  distcds 17306  TarskiGcstrkg 28435  Itvcitv 28441  LineGclng 28442  pInvGcmir 28660  ∟Gcrag 28701  ⟂Gcperpg 28703

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-tp 4631  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8014  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-oadd 8510  df-er 8745  df-map 8868  df-pm 8869  df-en 8986  df-dom 8987  df-sdom 8988  df-fin 8989  df-dju 9941  df-card 9979  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-nn 12267  df-2 12329  df-3 12330  df-n0 12527  df-xnn0 12600  df-z 12614  df-uz 12879  df-fz 13548  df-fzo 13695  df-hash 14370  df-word 14553  df-concat 14609  df-s1 14634  df-s2 14887  df-s3 14888  df-trkgc 28456  df-trkgb 28457  df-trkgcb 28458  df-trkg 28461  df-cgrg 28519  df-mir 28661  df-rag 28702  df-perpg 28704

This theorem is referenced by:  footexALT  28726  footexlem2  28728  colperpexlem3  28740  mideulem2  28742  lmimid  28802  hypcgrlem1  28807  hypcgrlem2  28808  trgcopyeulem  28813